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Theorem lautcnv 37298
Description: The converse of a lattice automorphism is a lattice automorphism. (Contributed by NM, 10-May-2013.)
Hypothesis
Ref Expression
lautcnv.i 𝐼 = (LAut‘𝐾)
Assertion
Ref Expression
lautcnv ((𝐾𝑉𝐹𝐼) → 𝐹𝐼)

Proof of Theorem lautcnv
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2824 . . . 4 (Base‘𝐾) = (Base‘𝐾)
2 lautcnv.i . . . 4 𝐼 = (LAut‘𝐾)
31, 2laut1o 37293 . . 3 ((𝐾𝑉𝐹𝐼) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
4 f1ocnv 6616 . . 3 (𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
53, 4syl 17 . 2 ((𝐾𝑉𝐹𝐼) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
6 eqid 2824 . . . 4 (le‘𝐾) = (le‘𝐾)
71, 6, 2lautcnvle 37297 . . 3 (((𝐾𝑉𝐹𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(le‘𝐾)𝑦 ↔ (𝐹𝑥)(le‘𝐾)(𝐹𝑦)))
87ralrimivva 3186 . 2 ((𝐾𝑉𝐹𝐼) → ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ (𝐹𝑥)(le‘𝐾)(𝐹𝑦)))
91, 6, 2islaut 37291 . . 3 (𝐾𝑉 → (𝐹𝐼 ↔ (𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ (𝐹𝑥)(le‘𝐾)(𝐹𝑦)))))
109adantr 484 . 2 ((𝐾𝑉𝐹𝐼) → (𝐹𝐼 ↔ (𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ (𝐹𝑥)(le‘𝐾)(𝐹𝑦)))))
115, 8, 10mpbir2and 712 1 ((𝐾𝑉𝐹𝐼) → 𝐹𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3133   class class class wbr 5053  ccnv 5542  1-1-ontowf1o 6343  cfv 6344  Basecbs 16481  lecple 16570  LAutclaut 37193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-ov 7149  df-oprab 7150  df-mpo 7151  df-map 8400  df-laut 37197
This theorem is referenced by:  ldilcnv  37323
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